To investigate the correlation between songs’ rank in the charts and their number of beats per minute.

This internal assessment is about the correlation between songs' positions in the charts and their beats per minute (BPM). Listening to music is one of my favourite activities, and I have always had a musical ear. I thought that this would be an excellent topic to investigate, as I am curious if these two things have any relationship.

I have decided to use charts lists from the United Kingdom (2012) and Colombia (2021) to allow for more representative and generalisable sample, in case it varies by region or has changed through time. To do this, I had to search online charts lists from the specified year and country, then find the BPM of each song online. Please refer to the References section for details on which online resources were consulted for data collection. I used UK-Charts for the list of the top 100 songs in the United Kingdom (2012), Top Charts for the list of the top 100 songs in Colombia (2021). To get the BPMs of each song, I used Spotify's API. Microsoft Excel was used a lot for keeping track, storing, and formatting the data. After collecting the data, the aim was to perform a selection of mathematical calculations to investigate any potential correlations there may be.

I have two hypotheses for this investigation: the Null Hypothesis (\(H_0\)) (there is no significant correlation between a song's position in the charts and its number of beats per minute), and the two-tailed Alternative Hypothesis (\(H_1\)), that the more highly ranked a song is in the charts, the more or fewer beats per minute it has.

The tools I used to analyse the data were all specialised online statistics calculators from https://www.socscistatistics.com/. I used this because of its high accuracy, speed, and overall convenience when compared to doing it by hand. For all of the undermentioned inferential statistics tests (except T-Test which used both together), I calculated twice: one for each data series.

Henceforth, the calculation using values from the UK 2012 data set shall be referred to as "Test A", and the calculations using data from the Colombia 2021 data set shall be referred to as "Test B".

The first test I performed on the data was a linear regression test.

X Values: Integers 1 to 100.
Y Values: BPMs of the UK data series (in the same order they appear in the table).
Results:
\(\hat{y}=-0.16865X+129.37903\)
Sum of X = 5050
Sum of Y = 12086.244
Mean X = 50.5
Mean Y = 120.8624
Sum of squares (SSX) = 83325
Sum of products (SP) = -14052.374
Regression Equation = \(\hat{y}=bX+a\)
b = SP/SSX = -14052.37/83325 = -0.16865
a = MY - bMX = 120.86 - (-0.17*50.5) = 129.37903
\(\hat{y}=-0.16865X+129.37903\)

X Values: Integers 1 to 100.
Y Values: BPMs of the Colombian data series (in the same order they appear in the table).
Results:
\(\hat{y}=-0.15638X+129.79066\)
Sum of X = 5050
Sum of Y = 12189.362
Mean X = 50.5
Mean Y = 121.8936
Sum of squares (SSX) = 83325
Sum of products (SP) = -13030.112
Regression Equation = \(\hat{y}=bX+a\)
b = SP/SSX = -13030.11/83325 = -0.15638
a = MY - bMX = 121.89 - (-0.16*50.5) = 129.79066
\(\hat{y}=-0.15638X+129.79066\)

Then, I calculated the Pearson correlation coefficient.

X Values: Integers 1 to 100.
Y Values: BPMs of the UK data series (in the same order they appear in the table).
Results: R = -0.2194
X Values
Σ = 5050
Mean = 50.5
\(\sum(X-Mx)^2=SSx=83325\)
Y Values
Σ = 12086.244
Mean = 120.862
\(\sum(Y-My)^2=SSy=49250.301\)
X and Y Combined
N = 100
\(\sum(X-Mx)(Y-My)=-14052.374\)
R Calculation
\(r=\sum((X-My)(Y-Mx))/\sqrt{(SSx)(SSy)}\)
\(r=-14052.374/\sqrt{(83325)(49250.301)}=-0.2194\)
Meta Numerics (cross-check)
r = -0.2194

X Values: Integers 1 to 100.
Y Values: BPMs of the Colombian data series (in the same order they appear in the table).
Results:
X Values
Σ = 5050
Mean = 50.5
\(\sum(X-Mx)^2=SSx=83325\)
Y Values
Σ = 12189.362
Mean = 121.894
\(\sum(Y-My)^2=SSy=106251.226\)
X and Y Combined
N = 100
\(\sum(X-Mx)(Y-My)=-13030.112\)
R Calculation
\(r=\sum((X-My)(Y-Mx))/\sqrt{(SSx)(SSy)}\)
\(r=-13030.112/\sqrt{(83325)(106251.226)}=-0.1385\)
Meta Numerics (cross-check)
r = -0.1385

I then used Spearman's Rho calculations to further analyse any correlation between the data.

X Values: Integers 1 to 100.
Y Values: BPMs of the UK data series (in the same order they appear in the table).
Results: \(r_s=-0.12515\), p(2-tailed) = 0.21474
X Ranks
Mean: 50.5
Standard Dev: 29.01
Y Ranks
Mean: 50.5
Standard Dev: 29.01
Combined
Covariance = -10428/99 = -105.33
R = -105.33/(29.01*29.01) = -0.125

X Values: Integers 1 to 100.
Y Values: BPMs of the Colombian data series (in the same order they appear in the table).
Results: \(r_s=-0.12515\), p(2-tailed) = 0.21474
X Ranks
Mean: 50.5
Standard Dev: 29.01
Y Ranks
Mean: 50.5
Standard Dev: 29.01
Combined
Covariance = -10428/99 = -105.33
R = -105.33/(29.01*29.01) = -0.125

Finally, I ran a T-Test between the BPMs of the British and Colombian charts lists.

Treatment 1: BPMs of the UK data series (in the same order they appear in the table).
Treatment 2: BPMs of the Colombian data series (in the same order they appear in the table).
Results: The t-value is -0.26019. The p-value is .79499. The result is not significant at p < .05

Treatment 1
\(N_1\): 100
\(df_1=N-1=100-1=99\)
\(M_1\): 120.86
\(SS_1\): 49250.3
\(s^2_1=SS_1/(N-1)=49250.3/(100-1)=497.48\)

Treatment 2
\(N_2\): 100
\(df_2=N-1=100-1=99\)
\(M_2\): 121.89
\(SS_2\): 106251.23
\(s^2_2=SS_2/(N-1)=106251.23/(100-1)=1073.24\)

T-value Calculation
\(s^2_p=((df_1/(df_1+df_2))*s^2_1)+((df_2/(df_2+df_2))*s^2_2)=((99/198)*497.48)+((99/198)*1073.24)=785.36\)
\(s^2_{M1}=s^2_p/N_1=785.36/100=7.85\)
\(s^2_{M2}=s^2_p/N_2=785.36/100=7.85\)
\(t=(M_1-M_2)/\sqrt{(s^2_{M1}+s^2_{M2})}=-1.03/\sqrt{15.71}=-0.26\)

I can conclude that the correlation between what I have been investigating in this IA is weak and negative. I therefore do not reject the null hypothesis. In all honesty, I was not expecting the results of this investigation to be particularly significant, due to the wide variety of musical styles with diverse tempos, all of which can be successful pieces in their own unique ways.

Top Charts. (2021, January 28). Colombia Top 100 Songs Chart | Top Charts. Retrieved from Top Charts: https://www.top-charts.com/songs/all-genres/colombia/total/2021-W19

UK-Charts. (n.d.). UK-Charts - Top 100 2012 (Full Year). Retrieved from UK-Charts: https://www.ukcharts.co.uk/index.php/charts/2010-s/154-2012/292-top-100-2012